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--- b:head_first_statistics:using_discrete_probability_distributions [2023/10/01 23:55] – [Theorem 4: Var(c) = 0] hkimscil
+++ b:head_first_statistics:using_discrete_probability_distributions [2023/10/04 10:29] – hkimscil
@@ Line 616: / Line 616: @@
 \begin{align*}
   &  E[(X + Y)^2] = E[X^2 + 2XY + Y^2] = E[X^2] + 2E[XY] + E[Y^2] \\
-- & [E(X + Y)]^2 = [E(X) + E(Y)]^2 = E(X)^2 + 2E(X)E(Y) + E(Y^2) \\
+- & [E(X + Y)]^2 = [E(X) + E(Y)]^2 = E(X)^2 + 2E(X)E(Y) + E(Y)^2 \\
 \end{align*}
@@ Line 624: / Line 624: @@
 Var[(X+Y)] =
   & E[X^2] & + & 2E[XY] & + & E[Y^2] \\
-- & E(X)^2 & - & 2E(X)E(Y) & - & E(Y^2) \\
+- & E(X)^2 & - & 2E(X)E(Y) & - & E(Y)^2 \\
   & Var[X] & + & 2 E[XY]-2E(X)E(Y) & + & Var[Y] \\
 \end{align*}
@@ Line 691: / Line 691: @@
 \end{align}
 ====== e.gs in R  ======
 R에서 이를 살펴보면
 <code>
-m <- 0
+# variance theorem 4-1, 4-2
-v <- 4
+# http://commres.net/wiki/variance_theorem
-n <- 10000
+# need a function, rnorm2
+rnorm2 <- function(n,mean,sd) { mean+sd*scale(rnorm(n)) }
+m <- 50
+v <- 100
+n <- 1000000
 set.seed(1)
 x1 <- rnorm2(n, m, sqrt(v))
@@ Line 718: / Line 722: @@
 v.12 <- var(x1 + x2)
 v.12
-## should be near 2*v
+######################################
+## v.12 should be near var(x1)+var(x2)
 ######################################
 ## 정확히 2*v가 아닌 이유는 x1, x2가
@@ Line 743: / Line 748: @@
 # var(2*x1) = 2^2 var(X1)
 v.11
 </code>